Question 1057729: An equilateral triangle with side 6cm is inscribed in a circle. Find its radius.
Answer by Boreal(15235)   (Show Source):
You can put this solution on YOUR website! Draw this.
While I don't do it here, the center to one vertex of the triangle is a radius.
That forms an isosceles triangle with the equal sides the radius and the long side one of the legs of the triangle. Drop a perpendicular from the center to the middle of the long side. That divides the triangle into two triangles with the common side as yet unknown.
Half of the long side is 3.
The distance from the center to the long side is half the hypotenuse or (1/2 r)^2, because this is a 30-60-90 triangle, and that is the opposite side.
Therefore, 3^2+(1/2 r)^2=r^2
r^2-(1/4)r^2=9
(3/4)r^2=9
r^2=12
r=2 sqrt (3)
This can be checked by noting one side of the equilateral triangle is the hypotenuse formed by a right triangle where one leg is half the side of the equilateral triangle and the radius plus the perpendicular to the side is another leg. The unknown side containing the radius is sqrt(27) or 3 sqrt (3).
The radius is at the median of the triangle, which is 2/3 of the way to the opposite side, making the radius (2/3) or 3 sqrt(3) or 2 sqrt (3).
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